NONLINEAR MATHEMATICAL MODEL TO DETERMINE THE STIFFNESS OF THE AUTOMOTIVE WHEEL BEARING Aleksandar Živković1, Milan Zeljković2, Slobodan Tabaković3 Summary: This paper presents a mathematical model with five degrees of freedom to determine the stiffness of the automotive wheel bearing. The mathematical model consists of two parts. The first part of the developed software solution to determine the contact stiffness for each rolling elements, while the second part on the basis of contact stiffness determines the stiffness of bearing based on the finite element method. Software solution is developed based on Hertz's contact load and John Harris of the quasi-static equilibrium equations. Contact bearing stiffness matrix is determined for different arrangements of bearings in the support. Mathematical model analyzes the influence of the cornering acceleration of the automotive wheel and clearance on the bearnig stiffness. The research in this paper shows that the cornering acceleration of the automotive wheel significantly affects the radial stiffness, while the clerances significantly influence the angular and axial stiffness. 1. INTRODUCTION In today's automotive industry is becoming more common developing the wheel bearings of vehicles based on the integration of each component axis ("Hub Unit Bearing - HUB") in order to reduce the weight and dimensions as well as to improve overall vehicle performance. Based on of present knowledge we can conclude that there are three directions of development of constructive solutions wheel bearing. One of the directions of the development of HUB unit bearing is advanced technology in making axle. This technology is based on a compact module HHM (Halfshaft Hub Module), which reduces the weight of the vehicle and increases the flexibility of the system for the suspension and steering systems. Another direction of development of constructive solutions wheel bearing vehicle is its integration with the brake drum. This solution provides a longer bearing life and rigidity while reducing weight, easy installation and lower costs. The third line of development of constructive solutions wheel bearing vehicles with integrated brake discs. Application of this solution is primarily expected in sports cars. 1
Dr, Aleksandar Živković, Novi Sad, University of Novi Sad, Faculty of tehnical science, (
[email protected]) Dr Milan Zeljković, Novi Sad, University of Novi Sad, Faculty of tehnical science, (
[email protected]) 3 Dr Slobodan Tabaković, Novi Sad, University of Novi Sad, Faculty of tehnical science, (
[email protected]) 2
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Aleksandar Živković, Milan Zeljković, Slobodan Tabaković
This paper analyzes the influence of lateral acceleration and clearance on stiffness of the HUB unit beraing second generation. The analysis was conducted using the developed mathematical model with five degrees of freedom. The mathematical model takes into account inertia forces, clearance in the bearing and bearing temperature elements. 2. MATHEMATICAL MODEL When defining the mathematical model used two coordinate system as shown in Figure 1 The first is a general coordinate system {x,y,z,θy,θz}T, which is associated with the degrees of freedom of rotating ring and is located in the center of the bearing. The second is a local coordinate system {r, x, }T, which defines the position of the center rotating raceways for each rolling element and is set in the center of curvature of the outer raceways of rolling bearings. It is assumed that the center of curvature of the inner raceways stationary and used as a fixed point when the outer ring is rotary. In the event that no action of external load centers of the raceways is located at a distance A, as shown in the Fig. 1. After effects of static load, the distance between the center of curvature of the inner and outer raceways A increases the size of the contact deformation u and s. The line between the load centers are collinear with distance A. However, the action of the centrifugal force and gyroscopic moment of rolling elements, due to the different angles of contact between the rolling body and the outer and inner raceways, the line of the load will is not be colinear with the distance between the centers. In Figure 2 shows the position of the center of rolling bodies and positions the center of curvature of the inner raceways with and without the action of centrifugal force and gyroscopic moment in rolling body.
Fig.1. Coordinate system of model with five degrees of freedom
Fig.2. Positions of ball center and raceway groove curvature centers at angular positions ψi with and without applied load In Figure 2, X1j, X2j, αij, αoj, ∆ij, ∆oj are the axial and radial components of the position of the center of the rolling body, angle of contact with the inner and outer 156
Nonlinear mathematical model to determine the stiffness of the automotive wheel bearing
raceways and rolling distance between the center of the body and the center of curvature of the inner and outer raceways. Bearing elements deform under the action of loads. On heating elements bearing leads to shrinkage of rolling bodies and expanding the rings. Due to these changes, there is a changing distance between the center of curvature of the inner and outer raceways and new position the center of the body. In case of considering the influence of clearance and temperature of this distance is defined as: ij fi 0.5 db ij Gr b (1) oj fo 0.5 db oj Gr b The distance between two positions of vectors in the ball bearing with angular contact is defined as: A1 j A sin p x z ri cos j y Ri sin j (2) A2 j A sin p y cos j z sin j
c1e y sin j z cos j o i ucent .
In the equations (1) i (2): fi=db/di, fo=db/do, db-rolling element diameter, di, dodiameter of the inner and outer raceways, δi, δo-deformations of the inner and outer raceways, Gr- positive clearance / negative clearance, εb- thermal expansion of the rolling element, e=2*ru*c2*tg(α0) - effective load center [2], constants c1 and c2 are related to raceways number and arrangement of the bearings in the support [2], εi, εothermal expansion of the inner and outer ring, ucent.-centrifugal expansion of the outer ring. Applying the Pythagorean theorem, according to Figure 2, we get the following equation of dispalcmenet:
A
1j
X 12j
A X l G X l G 0 X1 j
2
2
2j
2j
ij
ij
2
2j
o
r
2
b
0
(3)
2
oj
oj
r
b
The equation of the forces equilibrium on the roller element is: Qoj cos oj Qij cos ij Qoj sin oj Qij sin ij
M gj dk M gj dk
sin oj oj sin uj Fcj 0
cos oj ij cos ij 0
ij
oj
(4)
where: Qi i Qo- Hertz’s contact forces, Mg- gyroscopic moment, Fc- centrifugal force, λ- constant depending of the raceway control. Hertz's contact force between the inner raceways and rolling elements and the outer raceways and rolling elements are determined by the [3,4]: Ki i3/(i )2 Qi / o (i ) 3/ 2 K o o (i )
(5) Will get rolling with an outer or inner raceways depends on the fulfillment of the conditions set out in the table T.1 [1]. Table T1. Conditions in which there is pure rolling
E cos
Q
Outer ring
Qsj asj Esj cos uj sj Quj auj Euj
Inner ring
Quj auj
uj
uj
sj
sj a sj E sj
Under different conditions of rolling the body loses contact with the inner ring. In other words, the outer raceway generates the force of reaction between rolling 157
Aleksandar Živković, Milan Zeljković, Slobodan Tabaković
elements and the centripetal acceleration. As a result, the external angle of contact αoj j and gyroscopic moment becomes equal to 0 Contact force with the external ring Qoj in this case, is equal to the centrifugal force Fcj, respectively, with the external deformation path is doj=(Foj/Ko)2/3. To avoid loss of contact with the inner raceways must be satisfied the following condition [1]:
A12j A2 j f o 0.5 db K o2 / 3 Fcj2 / 3 fi 0.5 db 2
2
(6) Nonlinear motion equation (3) and the equilibrium equation (4) can be solved simultaneously using the Newton-Raphson's method of iteration to determine unknown X1j, X2j, δij i δoj. In the case of a rotary bearing outer ring, the forces and moments acting on the outer ring is calculated by the following equations:
Fx F y Fz M y M z
2
Z
a 1 i 1
M gj cos oj Qoj sin oj db M gj sin oj cos j Qoj cos oj db M gj cos sin sin Q oj oj oj j d b M gj ro Qoj sin oj cos oj f o M gj sin j db M gj cos oj f o M gj cos j ro Qoj sin oj d b
(7)
If the relative displacement bearing marked with km, where k = x, y, z, θy, θz and m= i, o, then the contact stiffness of bearing can be expressed by: KiL,k
Fku Z T Q j T j T j u k j 1 u j i
KoL,k
Fks Z T Q j T j T j s k j 1 u j o
(8) where [Tj] transformation matrix of the form: cos j sin j 0 sin j cos j T j 0 0 1 ru cos j ru cos j 0 0 0 sin j cos j (9) The total contact stiffness of bearing based on [5] calculated as stiffness inner and outer ring:
K L Ki Ko
(10) By changing the vehicle from linear to rotary motion, a significant action of moment and angular displacement of bearing around the Z axis. This makes for the HUB unit bearing vehicle is an important parameter and angular stiffness. However, the angular stiffness can not be determined using standard methods for the outer ring, which is significantly larger than the inner ring. The outer ring represents the hub and wheel vehicles. The most reliable way of determining the angular stiffness of the HUB unit bearing vehicles using finite element method. At the workshop drawings beraing, defined by a 3D model of the HUB unit bearing shown in Figure 3. Axial and radial 158
Nonlinear mathematical model to determine the stiffness of the automotive wheel bearing
effect of the rolling elements in rolling raceways (contact load) is defined in terms of contact pairs, for which particular attention was given to discretization raceways be identical throughout the volume. Based on the previously prepared data on loads and stiffness values for each rolling body in the processing part was established mathematical model (Fig. 3). Its solution is defined by the angular displacement of points A and B. Based on certain shifts in the points A and B, it is possible to determine the angular stiffness outer ring, and thus the whole bearing as [6]: u=(δA- δB) θ=tg-1(u/D) (11) kθ=M/θ where: u -angular displacement; A and B - displacement the points A and B, Mmoment, D-diameter outer ring acorrding to Figure 3
Fig.3. Mathematical model for determining the angular stiffness 3. RESULTS AND DISCUSSION Analysis stiffness of HUB unit bearng is made on the basis of reaction forces at the wheel of vehicles for a variety of conditions, lateral acceleration g = 0 to 0.8 and for different values of the clerances. 3.1 INFLUENCE OF LATERAL ACCELERATION OF BEARING STIFFNESS Due to the uneven distribution of the load and the contact angle on the rolling raceways, there is an unequal stiffness of raceways I and II. Since the raceway II all the rolling elements in the loading zone, it will appear at the raceway II greater rigidity. Figure 4 shows the stiffness in the radial direction of the raceways I and II with lateral acceleration G=0.4. The sum of these two stiffness gives the overall stiffness depending on the lateral acceleration. Figure 5 shows the modified total axial and radial stiffness depending on acceleration. The axial stiffness of the HUB unit bearing decreased with increasing lateral acceleration from 0 to 0.8 with 595 to 234 [N / μm], which caused a decrease in the radial stiffness of 642 to 323 [N / μm]. The biggest drop in the axial and radial stiffness after a lateral acceleration g = 0.4, after which it decreased by 56%. Since g = 0 to g = 0.4 stiffness is reduced by 9 %.
159
Radial stiffnes [N/ m]
Aleksandar Živković, Milan Zeljković, Slobodan Tabaković
120
40
100
35
80
30
60
25
40
20
20
15
0
0
100 200 300 Anular position []
400
10
0
100 200 300 Angular position [ ]
400
Fig.4. Changing the radial stiffness of each rolling body with G = 0.4::a) raceway I; b) raceway II according to Fig 3. 700 kxx kyy
Stiffness [N/ m]
600
500
400
300
200
0
0.2
0.4 Cornering acc. [-]
0.6
0.8
Fig.5. Changing the axial and radial stiffness depending on the lateral acceleration 7
500
Angular stiffness [Nm/rad]
Angular displacment [m/rad]
6 5 4 3 2
450
400
350
300
1 0
0
0.2 0.4 0.6 Cornering acc [-]
0.8
250
0
0.2 0.4 0.6 Cornering acc. [-]
0.8
Fig.6. Change of angular displacement and angular stiffness depending on the lateral acceleration 160
Nonlinear mathematical model to determine the stiffness of the automotive wheel bearing
Figure 6 shows the changes of angular displacements and angular stiffness depending on the lateral acceleration. Angular displacement of the HUB unit bearing growth by increasing lateral acceleration from 0 to 0.8 to 87%, with a reduction in angular stiffness by 46%. Angular stiffness to the lateral acceleration g = 0.4 decreases by 4%, followed by a sharp decline by 41%. 2.3 INFLUENCE OF AXIAL CLEARANCE OF BEARING STIFFNESS Another important parameter in the HUB unit bearing is clearance. In the analysis of the considered size axial clearance from -10 to +40 [μm] in steps of 5 [μm]. Considered the size of axial clearance are real-value used in the experimental tests, obtained by measuring the bearing during assembly. Figure 7 shows the change of axial and radial stiffness depending on the clearance. Axial displacement bearings increase by 26% by reducing negative clearance and increasing the clearance, and comes with a reduction in the axial stiffness by 12%. Graduating from the overlap clearance also leads to the increase of the radial displacement of 25%, which reduction of the radial stiffness of the HUB unit bearing for 6%. Angular stiffness and displacements are determined by the procedure described in the previous section. In Figure 8 show the change of angular stiffness depending on the clearance. Increasing clearance reduces the angular stiffness by 12%. From the above analysis it can be concluded that the lateral acceleration g = 0.4 has a greater influence on the radial stiffness, while clearance less influence than the angular and axial stiffness. 600
500
Stiffness [N/ m]
580
Angular siffness [Nm/ \rad]
kxx kyy
560 540 520 500 480 -10
0
10 20 Clerance [ m]
30
40
Fig.7. Changing the axial and radial stiffness depending on the clearance
480 460 440 420 -10
0
10 20 Clerance [m]
30
40
Fig.8. Changing the angular stiffness depending on the clearance
4. CONCLUSION For of the HUB unit bearring due to lower vehicle speed centrifugal force is small, so the difference between the contact load paths with rolling below 3%, so that the influence of inertial forces can be neglected. Due to the uneven distribution of the load and the contact angle on the rolling slopes, there is an unequal stiffness bearings. The axial stiffness of the HUB unit bearing decreased with increasing lateral acceleration. Another important parameter in the HUB unit bearing is axial clearance. 161
Aleksandar Živković, Milan Zeljković, Slobodan Tabaković
Number of rolling elements that transmit the load is directly dependent on the clerance. For higher values of axial clearance should be greater axial load, that all rolling elements involved in transferring loads. Reducing negative clerance and increasing the clerance, there is a reduction of the axial and radial stiffness. ACKNOWLEDGEMENTS In this paper some results of the project: Contemporary approaches to the development of special solutions related to bearing supports in mechanical engineering and medical prosthetics – TR 35025, carried out by the Faculty of Technical Sciences, University of Novi Sad, Serbia, are presented. The project is supported by Ministry of the education, science and technological development of the Republic of Serbia. REFERENCES [1] Cao, Y.(2006): Modeling of high-speed machine tools spindle system, Ph.D thesis, The University of British Columbia, 2006. [2] Gunduz, A (2012): Multi-dimensional stiffness characteristics of double row angular contact ball bearings and their role in influencing vibration modes, Doctoral Dissertation, The Ohio State University. [3] Harris, T. A., Michael, N. K. (2007): Rolling bearing analysis: Advanced Concepts of Bearing Technology, Fifth edition, Taylor & Francis Group [4] Harris, T. A., Michael, N. K. (2007): Rolling bearing analysis: Essential Concepts of Bearing Technology, Fifth edition, Taylor & Francis Group. [5] Li, H., Yung, C. S.(2004): Analysis of bearing configuration effects on high speed spindles using an integrated dynamic thermo-mechanical spindle model, International Journal of Machine Tools and Manufacture, Vol. 44, Pages 347–364. [6] Lee, I., Cho, Y., Kim, M., Jang, C., Lee, Y., Lee, S. (2012): Development of stiffness analysis program for automotive wheel bearing, 2012 Simulia Community Conference, pp 1-8. [7] Živković, A. (2013): Computer and experimental analysis of behavior ball bearings for special applications, Doctoral Dissertation, Faculty of technical science, University of Novi Sad, Serbia.
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